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The single pipeline is an or bitwise operator, so the expression self::ALLOW_ANONYMOUS_LIVE | self::ALLOW_ANONYMOUS_OFFLINE is the same as 1 | 2 and will evaluate to 3. Here is how the or bitwise operator works.

If the key is a not an integer then the key-value pair is left as is.

Source: https://github.com/craftcms/cms/blob/3.2.2/src/web/Controller.php#L75-L96

The single ampersand is an and bitwise operator, so the expression $test & $allowAnonymous will evaluate to true only if one or more of their bits intersect. Here is how the and bitwise operator works.

More on bitwise operators here.

Source: https://github.com/craftcms/cms/blob/3.2.2/src/web/Controller.php#L147-L176

I'm not sure what the specific issue is with your plugin, but as far as I can tell none of the behaviour from Craft 3.1.x has changed, rather than more specific values have been added.

The single pipeline | is a bitwise OR operator. It compares each bit (a binary digit of 0 or 1) in the integers and returns a new integer with a 1 wherever either numbers had a 1, and a 0 anywhere else (similar to how the || operator campares booleans). So the expression self::ALLOW_ANONYMOUS_LIVE | self::ALLOW_ANONYMOUS_OFFLINE is the same as 1 | 2 and will evaluate to 3. Here are some examples of how the | bitwise operator works:

Source: https://github.com/craftcms/cms/blob/3.2.2/src/web/Controller.php#L75-L96

If the key is a not an integer then the key-value pair is left as is.

Source: https://github.com/craftcms/cms/blob/3.2.2/src/web/Controller.php#L147-L176

The single ampersand & is a bitwise AND operator. It compares each bit in the integers and returns a new integer with a 1 wherever both numbers have a 1, and a 0 anywhere else (similar to how the && operator campares booleans). Here are some examples of how the & bitwise operator works:

By enforcing the condition $test & $allowAnonymous, we can check that $test (whether the site is live or offline) and $allowAnonymous (the permissions granted to the specific controller action) overlap.

More on bitwise operators here.

I'm not sure what the specific issue with your plugin is , but as far as I can tell the only thing that changed in Craft 3.2 was that the ALLOW_ANONYMOUS_OFFLINE constant was added, allowing for (but not requiring) more specific permissions on controller actions.

// Allow anonymous access to all actions only when the site is live
public $allowAnonymous = true;

// Allow anonymous access to the `preview` action only when the site is live
public $allowAnonymous = ['preview'];

// Allow anonymous access to the `preview` action in all situations
public $allowAnonymous = ['preview' => self::ALLOW_ANONYMOUS_LIVE | self::ALLOW_ANONYMOUS_OFFLINE];

// Allow anonymous access to the `preview` action only when the site is live
public $allowAnonymous = ['preview' => self::ALLOW_ANONYMOUS_LIVE];

// Allow anonymous access to the `preview` action only when the site is offline
public $allowAnonymous = ['preview' => self::ALLOW_ANONYMOUS_OFFLINE];

The init() method of the Craft base controller now normalizes the value as follows:

is the same as

// Enforce $allowAnonymous
$isLive = Craft::$app->getIsLive();
$test = $isLive ? self::ALLOW_ANONYMOUS_LIVE : self::ALLOW_ANONYMOUS_OFFLINE;

if (is_int($this->allowAnonymous)) {
    $allowAnonymous = $this->allowAnonymous;
} else {
    $allowAnonymous = $this->allowAnonymous[$action->id] ?? self::ALLOW_ANONYMOUS_NEVER;
}

if (!($test & $allowAnonymous)) {

The single ampersand is an AND bitwise operator, so the expression $test & $allowAnonymous will evaluate to true only if both bits evaluate to true (in other words neither are 0). More on bitwise operators here.

Source: https://github.com/craftcms/cms/blob/3.2.2/src/web/Controller.php#L147-L157

// Allow anonymous access to all actions only when the site is live
public $allowAnonymous = true;

// Allow anonymous access to the `preview` action only when the site is live
public $allowAnonymous = ['preview'];

// Allow anonymous access to the `preview` action only when the site is live
public $allowAnonymous = ['preview' => self::ALLOW_ANONYMOUS_LIVE];

// Allow anonymous access to the `preview` action only when the site is offline
public $allowAnonymous = ['preview' => self::ALLOW_ANONYMOUS_OFFLINE];

// Allow anonymous access to the `preview` action when the site is live or offline
public $allowAnonymous = ['preview' => self::ALLOW_ANONYMOUS_LIVE | self::ALLOW_ANONYMOUS_OFFLINE];

The single pipeline is an or bitwise operator, so the expression self::ALLOW_ANONYMOUS_LIVE | self::ALLOW_ANONYMOUS_OFFLINE is the same as 1 | 2 and will evaluate to 3. Here is how the or bitwise operator works.

( 0 = 0000) | ( 0 = 0000) => ( 0 = 0000)
( 0 = 0000) | ( 1 = 0001) => ( 1 = 0001)
( 0 = 0000) | ( 2 = 0010) => ( 2 = 0010)
( 1 = 0001) | ( 1 = 0001) => ( 1 = 0001)
( 1 = 0001) | ( 2 = 0010) => ( 3 = 0011)
( 2 = 0010) | ( 2 = 0010) => ( 2 = 0010)

The init() method of the Craft base controller now normalizes the value as follows:

equates to

// Enforce $allowAnonymous
$isLive = Craft::$app->getIsLive();
$test = $isLive ? self::ALLOW_ANONYMOUS_LIVE : self::ALLOW_ANONYMOUS_OFFLINE;

if (is_int($this->allowAnonymous)) {
    $allowAnonymous = $this->allowAnonymous;
} else {
    $allowAnonymous = $this->allowAnonymous[$action->id] ?? self::ALLOW_ANONYMOUS_NEVER;
}

if (!($test & $allowAnonymous)) {
    // Prevent access unless the user is logged in and has the required permissions to view the site
}

The single ampersand is an and bitwise operator, so the expression $test & $allowAnonymous will evaluate to true only if one or more of their bits intersect. Here is how the and bitwise operator works.

( 0 = 0000) & ( 0 = 0000) => ( 0 = 0000) => false
( 0 = 0000) & ( 1 = 0001) => ( 0 = 0000) => false
( 0 = 0000) & ( 2 = 0010) => ( 0 = 0000) => false
( 1 = 0001) & ( 1 = 0001) => ( 1 = 0001) => true
( 1 = 0001) & ( 2 = 0010) => ( 0 = 0000) => false
( 2 = 0010) & ( 2 = 0010) => ( 2 = 0010) => true
( 3 = 0011) & ( 1 = 0001) => ( 1 = 0001) => true
( 3 = 0011) & ( 2 = 0010) => ( 2 = 0010) => true
( 3 = 0011) & ( 3 = 0011) => ( 3 = 0011) => true

More on bitwise operators here.

Source: https://github.com/craftcms/cms/blob/3.2.2/src/web/Controller.php#L147-L176

// Allow anonymous access to all actions in all situations
public $allowAnonymous = true;

// Allow anonymous access to the `preview` action only when the site is live
public $allowAnonymous = ['preview'];

// Allow anonymous access to the `preview` action in all situations
public $allowAnonymous = ['preview' => self::ALLOW_ANONYMOUS_LIVE | self::ALLOW_ANONYMOUS_OFFLINE];

// Allow anonymous access to the `preview` action only when the site is live
public $allowAnonymous = ['preview' => self::ALLOW_ANONYMOUS_LIVE];

// Allow anonymous access to the `preview` action only when the site is offline
public $allowAnonymous = ['preview' => self::ALLOW_ANONYMOUS_OFFLINE];

The single ampersand is an AND bitwise operator, so the expression $test & $allowAnonymous will evaluate to true only if both bits evaluate to true (in other words neither are 0. More on bitwise operators here

// Allow anonymous access to all actions only when the site is live
public $allowAnonymous = true;

// Allow anonymous access to the `preview` action only when the site is live
public $allowAnonymous = ['preview'];

// Allow anonymous access to the `preview` action in all situations
public $allowAnonymous = ['preview' => self::ALLOW_ANONYMOUS_LIVE | self::ALLOW_ANONYMOUS_OFFLINE];

// Allow anonymous access to the `preview` action only when the site is live
public $allowAnonymous = ['preview' => self::ALLOW_ANONYMOUS_LIVE];

// Allow anonymous access to the `preview` action only when the site is offline
public $allowAnonymous = ['preview' => self::ALLOW_ANONYMOUS_OFFLINE];

The single ampersand is an AND bitwise operator, so the expression $test & $allowAnonymous will evaluate to true only if both bits evaluate to true (in other words neither are 0). More on bitwise operators here.